– Отец, а пространство бесконечно?
– Конечно, сын мой.
Investigating the laws of the Universe, scientists usually come to their conclusions using either induction (from particular to general) or deduction (from general to particular). The first methodology is invalid since it is all based on the principle of extrapolation which can be confirmed only by intuition. The second methodology is also quite arguable because it relies on some general laws which are clearly obtained through induction. In some sense, all the fundamental knowledge we have is derived by means of induction and therefore is unreasonable.
The area of knowledge which seems to be less doubtful is mathematics and logic particularly. In fact, logic is an essential part of any scientific method: both inductive and deductive approaches rely on it.
We can subdivide mathematics in applied math and pure math. Applied math is being developed to explain and predict reality. These two major functions of applied math are used in physics, computer science, economics, sociology and architecture. Pure math is more theoretical, however some of its chapters can be applied. Often pure math is done as an art, and great efforts are spent on things which are inapplicable to real life (for example, on studying flexible polyhedra). Applied math requires some purely mathematical results to proceed while pure math is completely independent.
How does applied math work? Dealing with some real-life problem, scientists begin with modelling reality. To make the model realistic enough, they make several assumptions based on real observations. Then from these inputs some conclusions are drawn by means of pure math. Finally, acquired results are interpreted or translated from the mathematical language of the model to the language of real life. But can we come to any true conclusions this way?
The problem is that models usually are irrelevant due to incompleteness of assumptions. Furthermore, these assumptions are always obtained through induction which validity is questionable. Therefore, according to the principles of logic, all interpretations of the model’s output are inappropriate since the model itself is wrong. The only reason models are useful is that some of them are of enough explanatory (or descriptive) power. They allow us to generalize our previous experience saying that it formed a certain pattern, but do not give us any confidence that this pattern will be hold further on. So, applied math is good at explanation but not at prediction.
Now what is pure math? It is the only piece of knowledge which is whole derived through deduction. It starts with definitions and axioms based on nothing. Then axioms give foundation to theorems and “practical” methods which help in solving purely mathematical problems. The key difference from the applied math is that there is no apparent connection with real life. Pure math is self-sufficient and the only way how it is linked with reality is that it is used in applied math as a tool. It allows scientists to derive conclusions which are all wrong from the epistemological point of view.
Pure math does not explain reality (if not being combined with applied math) and clearly does not predict it. Nevertheless, it seems to be the only truth we know. This great benefit is more than compensated by the abstractness which at the same time makes pure math truth. In fact, we have no guarantee that our mathematical reasoning is correct, and this is why it is better to say pure math is the only area of our knowledge that can be truth. As for other fields, the induction problem leaves just a miserable chance that we are right.
A good example of pure math is linear algebra. It operates with completely abstract notions and objects (such as n-dimensional linear vector space) which can model multiple real-life cases. For instance, linear algebra can be beautifully applied in physics. In combination with empiric Newtonian or Einsteinian laws it allows astronomers to calculate planets’ orbits and predict their movement. This does not mean that we know how planets revolve: there are many unaccounted factors in scientists’ models while all the accounted ones easily can be wrong due to the induction problem. However, we can be absolutely sure that the determinant of the product of two matrices is equal to the product of their determinants. That’s good news, isn’t it?